Examples 6.4.5(a):

Note that this function is continuous. The image of the interval
[-2, 1] is [1/5, 1]. In particular, the image of a compact set
is compact.

To find a closed set whose image is not closed we must consider
an unbounded set. Otherwise, a closed and bounded set is compact,
and since the image of a compact set under continuous functions
is compact, it is in particular closed again.

But if we look at the closed, unbounded set
[0, ),
we see that the image of that set under the above function is
the set (0, 1], which is not closed.